This post guides you step-by-step through how to use Descartes’ Rule of Signs, factoring, and long division to solve a fourth-degree polynomial. These techniques are foundational in algebra and essential for calculus students dealing with rational functions and polynomial division.

The Problem:

Solve
P(x) = 6x⁴ – 18x³ + 6x² – 30x + 36

Step 1: Use Descartes’ Rule of Signs

• Positive real zeros: Count sign changes in P(x) → 4 changes → 4 or 2 or 0 real positive roots.

• Negative real zeros: Count sign changes in P(–x) → 1 change → 1 or 0 negative real roots.

Conclusion: Either 4 real positive roots or 2 real + 2 complex roots.

Step 2: Factor Out a GCF

P(x) = 6(x⁴ – 3x³ + x² – 5x + 6)

Then test rational roots using ±factors of 6 over ±factors of 1 and 2:
Try x = 1, x = 3 → Both work.

Step 3: Long Division

Divide the polynomial by (x – 1)(x – 3) = x² – 4x + 3
The quotient is: x² + x + 2

Solve that quadratic:
x = (–1 ± √(1² – 4(1)(2))) / 2
→ x = –1/2 ± (i√7)/2 → 2 complex roots

Final Factored Form:

P(x) = 6(x – 1)(x – 3)(x² + x + 2)

Final Zeros:

• Rational Zeros: x = 1, x = 3

• Complex Zeros: x = –1/2 ± (i√7)/2

This lesson is pulled straight from the Ultimate Crash Course for STEM Majors, which includes everything you need for solving polynomials and preparing for calculus.

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